Cantor systems and quasi-isometry of groups

TL;DR

This paper explores the relationship between quasi-isometry of groups and topological orbit equivalence of Cantor minimal actions, introducing a new categorical framework and invariants for orbit equivalence relations.

Contribution

It establishes a correspondence between quasi-isometry of non-amenable groups and topological orbit equivalence of their Cantor minimal actions, and introduces a new categorical approach with invariants.

Findings

Non-amenable groups are quasi-isometric iff they have topologically orbit equivalent Cantor minimal actions.

Free groups of different rank admit topologically orbit equivalent actions, unlike in the measurable setting.

Introduces a representation of the orbit equivalence category, leading to new invariants.

Abstract

The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups if different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fundamental invariants of the orbit equivalence relation together with an ergodic invariant probability measure.